Common Errors of Post-Calculus Students
I. Common numerical/logical errors and misconceptions of post-calculus students:
(All variables refer to real numbers unless otherwise specified.)
1. a. If it is true that x < a, then it is not true that x ≤ a.
b. If x ≠ a, then it can’t be true that x ≤ a.
c. If x ≤ 4, then it is not true that x ≤ 2.
2. If one wants to prove that x ≤ a, and one has proven that x < a, one still has to consider the case
x = a.
3. a. The following statement is not true: For all real numbers x, x2 > -1.
b. The following statement is not true: For all real numbers x, x2 + 2 ≥ 1.
c. The following statement is not true: If x is a real number and y = x2 + 2, then y is in [1, ∞).
4. a. To prove that x is an element of the interval (3, ∞), one must prove both x > 3 and also x < ∞.
b. If x is an element of the interval (3, ∞), then x > 3 and x “goes on forever”.
5. If
y
/x > a, then it always follows that y > xa.
6. |(-x)| = x.
7. √(x2) = x.
8. a. x2 ≥ x.
8. b. If x > 0, then 1/x ≤ x.
9. If a < b, then a2 < b2 (and the converse).
10.
If xu = xv, then u = v.
11.
The only numbers in the interval [0, 2] are 0, 1, and 2.
(More specifically, if one needs to check something for all x in [0, 2], one needs only to check x = 0, 1, 2.)
12.
The set {1/n : n is a positive integer} (i.e., {x : there exists a positive integer n such that x = 1/n })
is the same as the set (0, 1].
13.
“Let m be the smallest number greater than 1.” (This is the one problem where you should prove something.)
14.
a. The equation x2 - 4x = y has no solutions because there are two variables and only one equation.
b. One can’t solve the equation x2 - 4x = y for x because y is unknown.
c. To solve the equation x2 - 4x = 7 for x,
I begin by factoring the left hand side: x(x - 4) = 7, so x = 7 or x - 4 = 7. …
15.
One can prove a statement by assuming that it is true and showing that it leads to a tautology (such as
1 = 1).E.g., if we want to prove the statement p, we can use the following logic: p  q  r “1=1”.
16.
a. If x2 < 1, then x < ± 1. (Is this an error? How?)
b. The following statement is not true: If x2 < 1, then x < 1.
II. Other occasional errors, not as common as those above:
1. If we know that k is a negative number, then k = - k. OR: If k is a negative number, then k = - a.
2. If x2 = 2, then x = |√2| .
Common Errors, Addendum
Some common errors which were listed on the sheet of errors for Project I:

|(-x)| = x.

√(x2) = x.

If a < b, then a2 < b2 (and the converse).

If y/x > a, then it always follows that y > xa.
QUESTION: What do these errors have in common? I.e., how could you explain in a general way the error
that one is making, an explanation that (at least partially) explains all the errors? Or, to look at it another
way, what common restriction could you make on the variables so that all these statements are true?